# Context-sensitive grammar

A context-sensitive grammar (CSG) is a formal grammar in which the left-hand sides and right-hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols. Context-sensitive grammars are more general than context-free grammars but still orderly enough to be parsed by a linear bounded automaton.

Noam Chomsky introduced context-sensitive grammars in the 1950s as a way to describe the syntax of natural language where it is indeed often the case that a word may or may not be appropriate in a certain place depending upon the context. A formal language that can be described by a context-sensitive grammar is called a context-sensitive language.

## Formal definition

A formal grammar G = (N, Σ, P, S), where N is a set of nonterminal symbols, Σ is a set of terminal symbols, P is a set of production rules, and S is the start symbol, is context-sensitive if all rules in P are of the form

αAβ → αγβ

where AN (i.e., A is a single nonterminal), α,β ∈ (N U Σ)* (i.e., α and β are strings of nonterminals and terminals) and γ ∈ (N U Σ)+ (i.e., γ is a nonempty string of nonterminals and terminals).

Some definitions of a context-sensitive grammar only require that for any production rule of the form u → v, the length of u shall be less than or equal to the length of v. This seemingly weaker requirement is claimed to be in fact equivalent.[1]

In addition, a rule of the form

S → λ

where λ represents the empty string and S does not appear on the right-hand side of any rule is permitted. The addition of the empty string allows the statement that the context sensitive languages are a proper superset of the context free languages, rather than having to make the weaker statement that all context free grammars with no →λ productions are also context sensitive grammars.

The name context-sensitive is explained by the α and β that form the context of A and determine whether A can be replaced with γ or not. This is different from a context-free grammar where the context of a nonterminal is not taken into consideration. (Indeed, every production of a context free grammar is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and/or nonterminals (w can be empty)).

If the possibility of adding the empty string to a language is added to the strings recognized by the noncontracting grammars (which can never include the empty string) then the languages in these two definitions are identical.

A formal language can be described by a context-sensitive grammar if and only if it is accepted by some linear bounded automaton.[2]

## Examples

1. SaSBC
2. SaBC
3. CBHB
4. HBHC
5. HCBC
6. aBab
7. bBbb
8. bCbc
9. cCcc

Rules 1 and 2 allow for blowing-up S to an(BC)n; rules 3 to 5 allow for successively exchanging each CB to BC (three rules are needed for that since a rule CBBC wouldn't fit into the scheme αAβ → αγβ); rules 6 to 9 allow for replacing a non-terminal B or C with its corresponding terminal b or c, provided it is in the right place. The generation chain for aaabbbccc is:

S
1 aSBC
1 aaSBCBC
2 aaaBCBCBC
3 aaaBHBCBC
4 aaaBHCCBC
5 aaaBBCCBC
3 aaaBBCHBC
4 aaaBBCHCC
5 aaaBBCBCC
3 aaaBBHBCC
4 aaaBBHCCC
5 aaaBBBCCC
6 aaabBBCCC
7 aaabbBCCC
7 aaabbbCCC
8 aaabbbcCC
9 aaabbbccC
9 aaabbbccc

More complicated grammars can be used to parse { anbncndn: n ≥ 1 }, and other languages with even more letters.

A context-sensitive grammar for the language $\{ a^{2^i} : i \geq 1 \}$ is constructed in Example 9.5 (p.224) of (Hopcroft, Ullman, 1979).

## Normal forms

Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form. "Equivalent" here means that the two grammars generate the same language. The normal form will not in general be context-sensitive, but will be a noncontracting grammar.

## Computational properties and uses

The decision problem that asks whether a certain string s belongs to the language of a certain context-sensitive grammar G, is PSPACE-complete. There are even some context-sensitive grammars whose fixed grammar recognition problem[clarification needed] is PSPACE-complete.

The emptiness problem for context-sensitive grammars (given a context-sensitive grammar G, is L(G)=∅ ?) is undecidable.

It has been shown that nearly all natural languages may in general be characterized by context-sensitive grammars, but the whole class of CSG's seems to be much bigger than natural languages[citation needed]. Worse yet, since the aforementioned decision problem for CSG's is PSPACE-complete, that makes them totally unworkable for practical use, as a polynomial-time algorithm for a PSPACE-complete problem would imply P=NP. Ongoing research on computational linguistics has focused on formulating other classes of languages that are "mildly context-sensitive" whose decision problems are feasible, such as tree-adjoining grammars, combinatory categorial grammars, coupled context-free languages, and linear context-free rewriting systems. The languages generated by these formalisms properly lie between the context-free and context-sensitive languages.